I would like to know if anybody knows any paper that is about the
evolution of entropy of the state estimation given a sequence of
observations in a hidden Markov chain before convergence to the
entropy rate of this Markov chain.

In other words, let $b$ a probability distribution over a finite set
of states. Given the probability of transition and emission of the
Markov chain, one can compute using the Bayes rules an estimation of
the state the chain is in. Do you know any work related to the
evolution of the entropy H(b), maybe related to entropy production
rate or some work like that ?

Some work on the capacity of a memoryless channel transmitting symbols
generated by a Markov source would be appreciated too :-)

I read some work from Mohammad Rezaeian (http://arxiv.org/abs/cs.IT/
0606114) on estimation entropy, but his work assumes convergence of
the Markov chain to the stationary regime.

Thank you very much,
CB.